Linear Systems: Continuous-Time Impulse Response Descriptions
An important input–output description of a linear continuous-time system is its impulse response, which is the response h(t, τ) to an impulse applied at time τ. In time-invariant systems that are also causal and at rest at time zero, the impulse response is h(t, 0) and its Laplace transform is the transfer function of the system. Expressions for h(t, τ) when the system is described by state-variable equations are also derived.
KeywordsLinear systems Continuous-time Time-invariant Time-varying Impulse response descriptions Transfer function descriptions
Connection to State-Variable Descriptions
Finally, it is easy to show that equivalent state-variable descriptions give rise to the same impulse responses.
The continuous-time impulse response is an external, input–output description of linear, continuous-time systems. When the system is time-invariant, the Laplace transform of the impulse response h(t, 0) (which is the output response at time t due to an impulse applied at time zero with initial conditions taken to be zero) is the transfer function of the system – another very common input–output description. The relationships with the state-variable descriptions are shown.
“Linear Systems: Continuous-Time, Time-Invariant, State Variable Descriptions,” Panos J. Antsaklis
“Linear Systems: Continuous-Time, Time-Varying, State Variable Descriptions,” Panos J. Antsaklis
External or input–output descriptions such as the impulse response and the transfer function (in the time-invariant case) are described in several textbooks below.
- Antsaklis PJ, Michel AN (2006) Linear systems. Birkhauser, BostonGoogle Scholar
- DeCarlo RA (1989) Linear systems. Prentice-Hall, Englewood CliffsGoogle Scholar
- Kailath T (1980) Linear systems. Prentice-Hall, Englewood CliffsGoogle Scholar
- Rugh WJ (1996) Linear systems theory, 2nd edn. Prentice-Hall, Englewood CliffsGoogle Scholar
- Sontag ED (1990) Mathematical control theory: deterministic finite dimensional systems. Texts in applied mathematics, vol 6. Springer, New YorkGoogle Scholar